# Experimental Verification of Real-Time Flow-Rate Estimations in a Tilting-Ladle-Type Automatic Pouring Machine

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## Abstract

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## 1. Introduction

## 2. Automatic Pouring Machine

## 3. Mathematical Models of Pouring Motion

#### 3.1. Motor Model for the Tilting Ladle

#### 3.2. The Pouring Process Model

#### 3.3. Load Cell Model

#### 3.4. Motor Model for Transferring the Ladle on the y- and z-Axes

#### 3.5. Synchronous Control for Transferring and Rotating the Ladle

## 4. Compensation of Measured Weight by Load Cell

## 5. Flow Rate Estimation by Decentralization of Kalman Filters

#### 5.1. The Design of Flow Rate Estimation

- Predict$$\begin{array}{ccc}\hfill {\widehat{x}}_{n}^{-}& =& f\left({\widehat{x}}_{n-1}\right),\hfill \\ \hfill {P}_{n}^{-}& =& {A}_{fn-1}{P}_{n}{A}_{fn-1}^{\mathrm{T}}+{Q}_{f},\hfill \end{array}$$
- Update$$\begin{array}{ccc}\hfill {G}_{n}& =& {P}_{n}^{-}{C}_{fn}^{\mathrm{T}}{({C}_{fn}{P}_{n}^{-}{C}_{fn}^{\mathrm{T}}+{R}_{f})}^{-1},\hfill \\ \hfill {\widehat{x}}_{n}& =& {\widehat{x}}_{n}^{-}+{G}_{n}\{{y}_{n}-\eta \left({\widehat{x}}_{n}^{-}\right)\},\hfill \\ \hfill {P}_{n}& =& (I-{G}_{n}{C}_{fn}^{\mathrm{T}}){P}_{n}^{-},\hfill \end{array}$$

#### 5.2. Simulations

## 6. Other Flow-Rate Estimation Methods for Comparison

#### 6.1. Flow-Rate Estimation by Differentiating Load Cell Data

#### 6.2. Flow-Rate Estimation Using a Visible Camera

## 7. Experimental Verifications

## 8. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Conflicts of Interest

## Abbreviations

CFD | Computational Fluid Dynamics |

SSKF | Steady State Kalman Filter |

EKF | Extended Kalman Filter |

DKFs | Decentralization of Kalman Filters |

DC | Direct Current |

IAE | Integral Absolute Error |

TV | Total Variation |

mTV | Modified Total Variation |

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**Figure 6.**Experimental results of the weight compensation system. (

**a**) Weight data measured by load cell; (

**b**) Angular velocity of ladle tilting; (

**c**) Acceleration on z-axis estimated by SSKF; (

**d**) Weight data transformed from estimated acceleration on z-axis using Equation (16).

**Figure 9.**Model parameters of ladle. (

**a**) Horizontal area on pouring mouth; (

**b**) Volume under pouring mouth; (

**c**) Derivative of horizontal area A with respect to angle $\theta $; (

**d**) Derivative of volume ${V}_{s}$ with respect to angle $\theta $.

**Figure 11.**Simulation results of flow-rate estimation with error of tilting angle +0 deg at the start of liquid outflow. (

**a**) Input command to motor for tilting ladle; (

**b**) Angular velocity of tilting ladle; (

**c**) Angle of tilting ladle; (

**d**) Liquid height on pouirng mouth; (

**e**) Flow rate of outflow liquid from ladle; (

**f**) Weight of outflow liquid from ladle.

**Figure 13.**Simulation results of flow rate estimation with error of tilting angle +3 deg at the start of liquid outflow. (

**a**) Input command to motor for tilting ladle; (

**b**) Angular velocity of tilting ladle; (

**c**) Angle of tilting ladle; (

**d**) Liquid height on pouirng mouth; (

**e**) Flow rate of outflow liquid from ladle; (

**f**) Weight of outflow liquid from ladle.

**Figure 17.**Experimental results using automatic pouring machine (tilting angle error at the start of liquid outflow: $+0$ deg). (

**a**) Input command to motor for tilting ladle; (

**b**) Angular velocity of tilting ladle; (

**c**) Angle of tilting ladle; (

**d**) Liquid height on pouring mouth; (

**e**) Flow rate of outflow liquid from ladle; (

**f**) Weight of outflow liquid from ladle.

**Figure 18.**Comparison of flow-rate estimations in experiments with tilting angle error at the start of liquid outflow: $+0$ deg. (

**a**) Estimated flow rate by EKF with covariance as shown in Equation (37); (

**b**) Estimated flow rate by EKF with covariance as shown in Equation (46); (

**c**) Estimated flow rate by differentiating weight of outflow liquid measured by load cell; (

**d**) Estimated flow rate by visible camera.

**Figure 19.**Experimental results using automatic pouring machine (tilting angle error at the start of liquid outflow: $+3$ deg). (

**a**) Input command to motor for tilting ladle; (

**b**) Angular velocity of tilting ladle; (

**c**) Angle of tilting ladle; (

**d**) Liquid height on pouring mouth; (

**e**) Flow rate of outflow liquid from ladle; (

**f**) Weight of outflow liquid from ladle.

**Figure 20.**Comparison of flow-rate estimations in experiments with tilting angle error at the start of liquid outflow: $+3$ deg. (

**a**) Estimated flow rate by EKF with covariance as shown in Equation (37); (

**b**) Estimated flow rate by EKF with covariance as shown in Equation (46); (

**c**) Estimated flow rate by differentiating weight of outflow liquid measured by load cell; (

**d**) Estimated flow rate by visible camera.

Estimation Method | Tilting Angle Error at Start | Tilting Angle Error at Start |
---|---|---|

to Liquid Outflow +0 deg | to Liquid Outflow +3 deg | |

DKFs with Covariance of Process Noise in Equation (37) | $4.277\times {10}^{-5}$ | $2.005\times {10}^{-4}$ |

DKFs with Covariance of Process Noise in Equation (46) | $6.236\times {10}^{-5}$ | $1.909\times {10}^{-4}$ |

Differentiating Load Cell Data | $2.431\times {10}^{-3}$ | $2.518\times {10}^{-3}$ |

Visible Camera | $1.420\times {10}^{-3}$ | $1.568\times {10}^{-3}$ |

Estimation Method | Tilting Angle Error at Start | Tilting Angle Error at Start |
---|---|---|

to Liquid Outflow +0 deg | to Liquid Outflow +3 deg | |

DKFs with Covariance of Process Noise in Equation (37) | $4.855\times {10}^{-4}$ | $4.746\times {10}^{-4}$ |

DKFs with Covariance of Process Noise in Equation (46) | $8.628\times {10}^{-4}$ | $8.357\times {10}^{-4}$ |

Differentiating Load Cell Data | $1.443\times {10}^{-1}$ | $1.486\times {10}^{-1}$ |

Visible Camera | $3.861\times {10}^{-2}$ | $4.006\times {10}^{-2}$ |

Estimation Method | Tilting Angle Error at Start | Tilting Angle Error at Start |
---|---|---|

to Liquid Outflow +0 deg | to Liquid Outflow +3 deg | |

DKFs with Covariance of Process Noise in Equation (37) | $8.549\times {10}^{-5}$ | $7.460\times {10}^{-5}$ |

DKFs with Covariance of Process Noise in Equation (46) | $4.628\times {10}^{-4}$ | $4.357\times {10}^{-4}$ |

Differentiating Load Cell Data | $1.439\times {10}^{-1}$ | $1.482\times {10}^{-1}$ |

Visible Camera | $3.821\times {10}^{-2}$ | $3.966\times {10}^{-2}$ |

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**MDPI and ACS Style**

Sueki, Y.; Noda, Y.
Experimental Verification of Real-Time Flow-Rate Estimations in a Tilting-Ladle-Type Automatic Pouring Machine. *Appl. Sci.* **2021**, *11*, 6701.
https://doi.org/10.3390/app11156701

**AMA Style**

Sueki Y, Noda Y.
Experimental Verification of Real-Time Flow-Rate Estimations in a Tilting-Ladle-Type Automatic Pouring Machine. *Applied Sciences*. 2021; 11(15):6701.
https://doi.org/10.3390/app11156701

**Chicago/Turabian Style**

Sueki, Yuta, and Yoshiyuki Noda.
2021. "Experimental Verification of Real-Time Flow-Rate Estimations in a Tilting-Ladle-Type Automatic Pouring Machine" *Applied Sciences* 11, no. 15: 6701.
https://doi.org/10.3390/app11156701